Properties

Label 1-435-435.92-r0-0-0
Degree $1$
Conductor $435$
Sign $0.640 - 0.768i$
Analytic cond. $2.02013$
Root an. cond. $2.02013$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.433 − 0.900i)7-s + (−0.781 − 0.623i)8-s + (0.623 + 0.781i)11-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + i·17-s + (−0.900 − 0.433i)19-s + (−0.433 − 0.900i)22-s + (0.974 − 0.222i)23-s + (−0.900 + 0.433i)26-s − i·28-s + (0.222 − 0.974i)31-s + (−0.433 − 0.900i)32-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.433 − 0.900i)7-s + (−0.781 − 0.623i)8-s + (0.623 + 0.781i)11-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + i·17-s + (−0.900 − 0.433i)19-s + (−0.433 − 0.900i)22-s + (0.974 − 0.222i)23-s + (−0.900 + 0.433i)26-s − i·28-s + (0.222 − 0.974i)31-s + (−0.433 − 0.900i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.640 - 0.768i$
Analytic conductor: \(2.02013\)
Root analytic conductor: \(2.02013\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 435,\ (0:\ ),\ 0.640 - 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7527608651 - 0.3526831932i\)
\(L(\frac12)\) \(\approx\) \(0.7527608651 - 0.3526831932i\)
\(L(1)\) \(\approx\) \(0.7166897083 - 0.1506614239i\)
\(L(1)\) \(\approx\) \(0.7166897083 - 0.1506614239i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.974 - 0.222i)T \)
7 \( 1 + (-0.433 - 0.900i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.900 - 0.433i)T \)
23 \( 1 + (0.974 - 0.222i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (-0.781 - 0.623i)T \)
41 \( 1 + T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (-0.974 - 0.222i)T \)
59 \( 1 + T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (0.781 + 0.623i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.974 - 0.222i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.433 - 0.900i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.433 + 0.900i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.552643570580290426350582504557, −23.52532663557753498129726290844, −22.56765082843238356166753733257, −21.36763060719342279140428076339, −20.85017297301185059229305727284, −19.49011754606818313289132707968, −19.03531668452509759958835531740, −18.31218003811867657671689328824, −17.30701661891363686450545406923, −16.33234101843038448372795122771, −15.852314293036036677429722771798, −14.79933780253024086541637538648, −13.88163221856556214672469831378, −12.54761193566680263088253562526, −11.57632148337980138612116571297, −10.92075590615716765397169794896, −9.65208186672376060312899397142, −8.92202138800287346531944620534, −8.32033990764606632778242748983, −6.87653024080697694204143660329, −6.27058088011239265269676164216, −5.23076799063756900203951780225, −3.5106224238422692719337282242, −2.4194826787042186946062536868, −1.12542762018309472814904773525, 0.79627466061113430516206843741, 2.00795215767699611542203826430, 3.37714608145701415991982666528, 4.28412451921200782044543449544, 6.09272538845076783089337360954, 6.85440333719254840087946533013, 7.79384652316446613816996302005, 8.78673734750676768331817767194, 9.69855785910884176311873365386, 10.61634672917350869499866716951, 11.18037667659044507170194040552, 12.58834992378123981139934631747, 13.07070951402184691670036796019, 14.562544705916018345313451464016, 15.44014967178231048949156638143, 16.37098670966437023501624886936, 17.36369998171824844125907008849, 17.6071951889328707373178451900, 19.08018483409347525408924280797, 19.47108331353790553929609672189, 20.49053254708135733004318728620, 21.01140957942976058732486304064, 22.28203368421220956610933358347, 23.126508594242420879223986380973, 24.08671757567564092013025180337

Graph of the $Z$-function along the critical line